#### Volume 13, issue 2 (2009)

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Flats and the flat torus theorem in systolic spaces

### Tomasz Elsner

Geometry & Topology 13 (2009) 661–698
##### Abstract

We prove the Systolic Flat Torus Theorem, which completes the list of basic properties that are simultaneously true for systolic geometry and $CAT\left(0\right)$ geometry.

We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries.

##### Keywords
systolic complex, systolic group, minimal surface, flat, flat torus
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 53C21